共查询到20条相似文献,搜索用时 171 毫秒
1.
In this paper, we study Perelman’s W{{\mathcal W}} -entropy formula for the heat equation associated with the Witten Laplacian on complete Riemannian manifolds via the Bakry–Emery
Ricci curvature. Under the assumption that the m-dimensional Bakry–Emery Ricci curvature is bounded from below, we prove an analogue of Perelman’s and Ni’s entropy formula
for the W{\mathcal{W}} -entropy of the heat kernel of the Witten Laplacian on complete Riemannian manifolds with some natural geometric conditions.
In particular, we prove a monotonicity theorem and a rigidity theorem for the W{{\mathcal W}} -entropy on complete Riemannian manifolds with non-negative m-dimensional Bakry–Emery Ricci curvature. Moreover, we give a probabilistic interpretation of the W{\mathcal{W}} -entropy for the heat equation of the Witten Laplacian on complete Riemannian manifolds, and for the Ricci flow on compact
Riemannian manifolds. 相似文献
2.
Xiang-Dong Li 《Journal of Geometric Analysis》2010,20(2):354-387
We prove some Sobolev inequalities on differential forms over a class of complete non-compact Riemannian manifolds with suitable
geometric conditions. Moreover, we establish some L
p,q
-estimates and existence theorems of the Cartan-De Rham equation and the Hodge systems. As applications, we prove some vanishing
theorems of the L
p,q
-cohomology and prove the L
q
-solvability of the nonlinear p-Laplace equation on forms on complete non-compact Riemannian manifolds with suitable geometric conditions. 相似文献
3.
Gabriela Araujo-Pardo Isabel Hubard Deborah Oliveros Egon Schulte 《Israel Journal of Mathematics》2013,195(2):647-675
The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds. 相似文献
4.
Mary R. Sandoval 《偏微分方程通讯》2013,38(10):1818-1846
Given a compact boundaryless Riemannian manifold upon which a compact Lie group G acts by isometries, recall that the G-invariant Laplacian is the restriction of the ordinary Laplacian on functions to the space of functions which are constant along the orbits of the action. By considering the wave trace of the invariant Laplacian and the connection between G manifolds and Riemannian foliations, invariants of the spectrum of the G-invariant Laplacian can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the principal orbits and contained in the open dense set of principal orbits are associated to the singularities of the wave trace of the G-invariant spectrum. If the action admits finite orbits, then the invariants also include the lengths of certain geodesics arcs connecting the finite orbit to itself. Under additional hypotheses, we obtain partial wave traces. As an application, a partial trace formula for Riemannian foliations with bundle-like metrics is also presented, as well as several special cases where better results are available. 相似文献
5.
Hajime Urakawa 《Geometriae Dedicata》1999,74(1):95-112
We give a graph theoretic analogue of Cheng's eigenvalue comparison theorems for the Laplacian of complete Riemannian manifolds. As its applications, we determine the infimum of the (essential) spectrum of the discrete Laplacian for infinite graphs. 相似文献
6.
Following Mark Kac, it is said that a geometric property of a compact Riemannian manifold can be heard if it can be determined from the eigenvalue spectrum of the associated Laplace operator on functions. On the contrary, D’Atri
spaces, manifolds of type A{\mathcal{A}}, probabilistic commutative spaces,
\mathfrakC{\mathfrak{C}}-spaces,
\mathfrakTC{\mathfrak{TC}}-spaces, and
\mathfrakGC{\mathfrak{GC}}-spaces have been studied by many authors as symmetric-like Riemannian manifolds. In this article, we prove that for closed
Riemannian manifolds, none of the properties just mentioned can be heard. Another class of interest is the class of weakly
symmetric manifolds. We consider the local version of this property and show that weak local symmetry is another inaudible
property of Riemannian manifolds. 相似文献
7.
I. Kath 《Transformation Groups》2000,5(2):157-179
The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors. We will prove that under a certain additional condition a dual pseudo-Riemannian space (G/H, g) of a compact Riemannian homogeneous space (G/H, g) with homogeneousSpin-structure admits a homogeneousSpin
+-structure and theG_invariant Killing spinors on (G/H, g) correspond toG-invariant Killing spinors on (G/H, g). We can ensure that in most cases the hypothesis onG-invariance is satisfied. 相似文献
8.
Forman 《Discrete and Computational Geometry》2003,29(3):323-374
Abstract. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds.
We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions
to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing
the properties of the combinatorial Ricci curvature with those of its Riemannian avatar. 相似文献
9.
Forman 《Discrete and Computational Geometry》2008,29(3):323-374
Abstract. In this paper we present a new notion of curvature for cell complexes. For each p , we define a p th combinatorial curvature function, which assigns a number to each p -cell of the complex. The curvature of a p -cell depends only on the relationships between the cell and its neighbors. In the case that p=1 , the curvature function appears to play the role for cell complexes that Ricci curvature plays for Riemannian manifolds.
We begin by deriving a combinatorial analogue of Bochner's theorems, which demonstrate that there are topological restrictions
to a space having a cell decomposition with everywhere positive curvature. Much of the rest of this paper is devoted to comparing
the properties of the combinatorial Ricci curvature with those of its Riemannian avatar. 相似文献
10.
In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle TM of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to TM, such that the vertical and horizontal distributions VTM, HTM are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging VTM, HTM (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on TM such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on TM. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1].Partially supported by the Grant 100/2003, MECT-CNCSIS, România. 相似文献
11.
Andrea Mondino 《Journal of Geometric Analysis》2013,23(2):764-811
The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds (M,g)) is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds (?3,g ? )—where g ? is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds (M,g) of dimension three. 相似文献
12.
Olaf Post 《Annales Henri Poincare》2006,7(5):933-973
We consider a family of non-compact manifolds Xε (“graph-like manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian
and the generalized Neumann (Kirchhoff) Laplacian
on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions.
As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need
to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in
the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence
results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also
in other geometric situations.
Communicated by Claude Alain Pillet
Submitted: December 21, 2005 Accepted: January 30, 2006 相似文献
13.
Daniel A. Ramras 《Archiv der Mathematik》2011,96(6):589-599
We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We apply this result to the Morse strata of the Yang-Mills functional over a closed
surface. The resulting neighborhoods play an important role in calculations of gauge-equivariant cohomology for moduli spaces
of flat connections over non-orientable surfaces. 相似文献
14.
Let (M,F) be a Finsler manifold, and let TM 0 be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM 0,G) and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces. 相似文献
15.
Mark Harmer 《Journal of Geometric Analysis》2009,19(2):358-372
We propose geometric conditions equivalent to the discreteness of the spectrum of the Laplacian on a class of Riemannian manifolds
with ends close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum
and stochastic incompleteness.
相似文献
16.
Lyle Noakes 《Advances in Computational Mathematics》2002,17(4):385-395
Riemannian quadratics are C
1 curves on Riemannian manifolds, obtained by performing the quadratic recursive deCastlejeau algorithm in a Riemannian setting. They are of interest for interpolation problems in Riemannian manifolds, such as trajectory-planning for rigid body motion. Some interpolation properties of Riemannian quadratics are analysed when the ambient manifold is a sphere or projective space, with the usual Riemannian metrics. 相似文献
17.
Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary
and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce
miscellaneous related results.
The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006).
The work was done when Z. Chen was on sabbatical in China. 相似文献
18.
Zhiren Jin 《Inventiones Mathematicae》1992,108(1):1-10
Summary We prove several Liouville theorems for harmonic maps between certain classes of Riemannian manifolds. In particular, the results can be applied to harmonic maps from the Euclidean space (R
m
,g
0) to a large class of Riemannian manifolds. Our assumptions on the harmonic maps concern the asymptotic behavior of the maps at .Oblatum 28-XII-1990 & 11-II-1991Supported by NSF grant DMS-8610730 相似文献
19.
Michael Carr 《Topology and its Applications》2006,153(12):2155-2168
Given a graph Γ, we construct a simple, convex polytope, dubbed graph-associahedra, whose face poset is based on the connected subgraphs of Γ. This provides a natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron. Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric and combinatorial properties of the complex as well as of the polyhedra are given. These spaces are natural generalizations of the Deligne-Knudsen-Mumford compactification of the real moduli space of curves. 相似文献
20.
Yong Hah Lee 《Potential Analysis》2005,23(1):83-97
In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds.
This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation. 相似文献